Casazza et al. [1]} studied \(\mathcal {X}_d\) -frames and \(\mathcal {X}_d\) -Bessel sequences in Banach spaces. Stoeva [2]} gave some perturbation results for \(\mathcal {X}_d\) -frames and atomic decompositions. Kaushik and Sharma [3]} studied approximative atomic decompositions in Banach spaces. For further studies related to approximative frame one may refer [4]}, [5]}, [6]}. Gavruta [7]} introduced and studied atomic system for an operator \(K\)  and the notion of \(K\) -frame in a Hilbert space. Xiao et al. [8]} discussed relationship between \(K\) -frames and ordinary frames in Hilbert spaces. Poumai and Jahan [9]} introduced K-atomic decompositions in Banach spaces.
